Landis-Oleinik Conjecture in the Exterior Domain

TL;DR

This paper proves the Landis-Oleinik conjecture for parabolic equations in exterior domains, showing that solutions with rapid decay must be identically zero, under certain coefficient conditions, advancing previous results.

Contribution

The authors establish the conjecture under weaker assumptions on coefficients, extending the class of equations for which the decay implies triviality.

Findings

Proved the Landis-Oleinik conjecture in exterior domains.

Extended previous results by relaxing coefficient conditions.

Demonstrated decay implies trivial solutions under new assumptions.

Abstract

In 1974, Landis and Oleinik conjectured that if a bounded solution of a parabolic equation decays fast at a time, then the solution must vanish identically before that time, provided the coefficients of the equation satisfy appropriate conditions at infinity. We prove this conjecture under some reasonable assumptions on the coefficients which improved the earlier results.